Maschke's theorem

Formulations

Maschke's theorem addresses the question: when is a general (finite-dimensional) representation built from irreducible subrepresentations using the direct sum operation? This question (and its answer) are formulated differently for different perspectives on group representation theory.

Group-theoretic

Maschke's theorem is commonly formulated as a corollary to the following result:

Then the corollary is

The vector space of complex-valued class functions of a group G has a natural G-invariant inner product structure, described in the article Schur orthogonality relations. Maschke's theorem was originally proved for the case of representations over \Complex by constructing U as the orthogonal complement of W under this inner product.

Module-theoretic

One of the approaches to representations of finite groups is through module theory. Representations of a group G are replaced by modules over its group algebra K[G] (to be precise, there is an isomorphism of categories between K[G]\text{-Mod} and \operatorname{Rep}_{G}, the category of representations of G). Irreducible representations correspond to simple modules. In the module-theoretic language, Maschke's theorem asks: is an arbitrary module semisimple? In this context, the theorem can be reformulated as follows:

The importance of this result stems from the well developed theory of semisimple rings, in particular, their classification as given by the Wedderburn–Artin theorem. When K is the field of complex numbers, this shows that the algebra K[G] is a product of several copies of complex matrix algebras, one for each irreducible representation. If the field K has characteristic zero, but is not algebraically closed, for example if K is the field of real or rational numbers, then a somewhat more complicated statement holds: the group algebra K[G] is a product of matrix algebras over division rings over K. The summands correspond to irreducible representations of G over K.

Category-theoretic

Reformulated in the language of semi-simple categories, Maschke's theorem states

Proofs

Group-theoretic

Let U be a subspace of V complement of W. Let p_0 : V \to W be the projection function, i.e., p_0(w + u) = w for any u \in U, w \in W.

Define p(x) = \frac{1}{#G} \sum_{g \in G} g \cdot p_0 \cdot g^{-1} (x), where g \cdot p_0 \cdot g^{-1} is an abbreviation of \rho_W{g} \cdot p_0 \cdot \rho_V{g^{-1}}, with \rho_W{g}, \rho_V{g^{-1}} being the representation of G on W and V. Then, \ker p is preserved by G under representation \rho_V: for any w' \in \ker p, h \in G, \begin{align} p(hw') &= h \cdot h^{-1} \frac{1}{#G} \sum_{g \in G} g \cdot p_0 \cdot g^{-1} (hw') \ &= h \cdot \frac{1}{#G} \sum_{g \in G} (h^{-1} \cdot g) \cdot p_0 \cdot (g^{-1} h) w' \ &= h \cdot \frac{1}{#G} \sum_{g \in G} g \cdot p_0 \cdot g^{-1} w' \ &= h \cdot p(w') \ &= 0 \end{align}

so w' \in \ker p implies that hw' \in \ker p . So the restriction of \rho_V on \ker p is also a representation.

By the definition of p, for any w \in W, p(w) = w, so W \cap \ker\ p = {0}, and for any v \in V, p(p(v)) = p(v). Thus, p(v-p(v)) = 0, and v - p(v) \in \ker p. Therefore, V = W \oplus \ker p.

Module-theoretic

Let V be a K[G]-submodule. We will prove that V is a direct summand. Let π be any K-linear projection of K[G] onto V. Consider the map \begin{cases} \varphi:K[G]\to V \ \varphi:x \mapsto \frac{1}{#G}\sum_{s \in G} s\cdot \pi(s^{-1} \cdot x) \end{cases}

Then φ is again a projection: it is clearly K-linear, maps K[G] to V, and induces the identity on V (therefore, maps K[G] onto V). Moreover we have

\begin{align} \varphi(t\cdot x) &= \frac{1}{#G}\sum_{s \in G} s\cdot \pi(s^{-1}\cdot t\cdot x)\ &= \frac{1}{#G}\sum_{u \in G} t\cdot u\cdot \pi(u^{-1}\cdot x)\ &= t\cdot\varphi(x), \end{align}

so φ is in fact K[G]-linear. By the splitting lemma, K[G]=V \oplus \ker \varphi. This proves that every submodule is a direct summand, that is, K[G] is semisimple.

Converse statement

The above proof depends on the fact that #G is invertible in K. This might lead one to ask if the converse of Maschke's theorem also holds: if the characteristic of K divides the order of G, does it follow that K[G] is not semisimple? The answer is yes.

Proof. For x = \sum\lambda_g g\in K[G] define \epsilon(x) = \sum\lambda_g. Let I=\ker\epsilon. Then I is a K[G]-submodule. We will prove that for every nontrivial submodule V of K[G], I \cap V \neq 0. Let V be given, and let v=\sum\mu_gg be any nonzero element of V. If \epsilon(v)=0, the claim is immediate. Otherwise, let s = \sum 1 g. Then \epsilon(s) = #G \cdot 1 = 0 so s \in I and sv = \left(\sum1g\right)!\left(\sum\mu_gg\right) = \sum\epsilon(v)g = \epsilon(v)s

so that sv is a nonzero element of both I and V. This proves V is not a direct complement of I for all V, so K[G] is not semisimple.

Non-examples

The theorem can not apply to the case where G is infinite, or when the field K has characteristics dividing #G. For example,

• Consider the infinite group \mathbb{Z} and the representation \rho: \mathbb{Z} \to \mathrm{GL}_2(\Complex) defined by \rho(n) = \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix}^n = \begin{bmatrix} 1 & n \ 0 & 1 \end{bmatrix}. Let W = \Complex \cdot \begin{bmatrix} 1 \ 0 \end{bmatrix}, a 1-dimensional subspace of \Complex^2 spanned by \begin{bmatrix} 1 \ 0 \end{bmatrix}. Then the restriction of \rho on W is a trivial subrepresentation of \mathbb{Z} . However, there's no U such that both W, U are subrepresentations of \mathbb{Z} and \Complex^2 = W \oplus U: any such U needs to be 1-dimensional, but any 1-dimensional subspace preserved by \rho has to be spanned by an eigenvector for \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix}, and the only eigenvector for that is \begin{bmatrix} 1 \ 0 \end{bmatrix}.
• Consider a prime p, and the group \mathbb{Z}/p\mathbb{Z}, field K = \mathbb{F}_p, and the representation \rho: \mathbb{Z}/p\mathbb{Z} \to \mathrm{GL}_2(\mathbb{F}_p) defined by \rho(n) = \begin{bmatrix} 1 & n \ 0 & 1 \end{bmatrix}. Simple calculations show that there is only one eigenvector for \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} here, so by the same argument, the 1-dimensional subrepresentation of \mathbb{Z}/p\mathbb{Z} is unique, and \mathbb{Z}/p\mathbb{Z} cannot be decomposed into the direct sum of two 1-dimensional subrepresentations.

Notes

References

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References

1. Maschke, Heinrich. (1898-07-22). "Ueber den arithmetischen Charakter der Coefficienten der Substitutionen endlicher linearer Substitutionsgruppen". Math. Ann..
2. Maschke, Heinrich. (1899-07-27). "Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind". Math. Ann..
3. "Heinrich Maschke".
4. It follows that every module over K[G] is a semisimple module.
5. The converse statement also holds: if the characteristic of the field divides the order of the group (the ''modular case''), then the group algebra is not semisimple.
6. The number of the summands can be computed, and turns out to be equal to the number of the [[conjugacy class]]es of the group.
7. One must be careful, since a representation may decompose differently over different fields: a representation may be irreducible over the real numbers but not over the complex numbers.